Complex analytic variety: Difference between revisions
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*{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Géométrie algébrique et géométrie analytique | url=http://www.numdam.org/numdam-bin/item?id=AIF_1956__6__1_0 | mr=0082175 | year=1956 | journal=[[Annales de l'Institut Fourier]] | issn=0373-0956 | volume=6 | pages=1–42 | doi=10.5802/aif.59| doi-access=free }} |
*{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Géométrie algébrique et géométrie analytique | url=http://www.numdam.org/numdam-bin/item?id=AIF_1956__6__1_0 | mr=0082175 | year=1956 | journal=[[Annales de l'Institut Fourier]] | issn=0373-0956 | volume=6 | pages=1–42 | doi=10.5802/aif.59| doi-access=free }} |
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* {{cite book |arxiv=math/0206203|last1=Grothendieck|first1=Alexander|last2=Raynaud|first2=Michele|title=Revêtements étales et groupe fondamental (SGA 1)|chapter =Revêtements étales et groupe fondamental§XII. Géométrie algébrique et géométrie analytique|year=2002|isbn=978-2-85629-141-2|chapter-url=https://link.springer.com/chapter/10.1007%2FBFb0058667|doi=10.1007/BFb0058656|language=fr}} |
* {{cite book |arxiv=math/0206203|last1=Grothendieck|first1=Alexander|last2=Raynaud|first2=Michele|title=Revêtements étales et groupe fondamental (SGA 1)|chapter =Revêtements étales et groupe fondamental§XII. Géométrie algébrique et géométrie analytique|year=2002|isbn=978-2-85629-141-2|chapter-url=https://link.springer.com/chapter/10.1007%2FBFb0058667|doi=10.1007/BFb0058656|language=fr}} |
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*{{cite book |isbn=978-4-431-49822-3}} |
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Revision as of 04:34, 11 December 2021
In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety or complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic varieties are locally ringed spaces which are locally isomorphic to local model spaces, where a local model space is an open subset of the vanishing locus of a finite set of holomorphic functions.
Definition
Denote the constant sheaf on a topological space with value by . A -space is a locally ringed space whose structure sheaf is an algebra over .
Choose an open subset of some complex affine space , and fix finitely many holomorphic functions in . Let be the common vanishing locus of these holomorphic functions, that is, . Define a sheaf of rings on by letting be the restriction to of , where is the sheaf of holomorphic functions on . Then the locally ringed -space is a local model space.
A complex analytic variety is a locally ringed -space which is locally isomorphic to a local model space.
Morphisms of complex analytic varieties are defined to be morphisms of the underlying locally ringed spaces, they are also called holomorphic maps.
See also
References
- Grauert, Hans, and Reinhold Remmert. "Coherent analytic sheaves." Vol. 265. Springer Science & Business Media, 2012.
- Grauert, Peternell, and Remmert, Encyclopaedia of Mathematical Sciences 74: Several Complex Variables VII
- Cartan, H.; Bruhat, F.; Cerf, Jean.; Dolbeault, P.; Frenkel, Jean.; Hervé, Michel; Malatian.; Serre, J-P. "Séminaire Henri Cartan, Tome 4 (1951-1952)". (no.10-13)
- Serre, Jean-Pierre (1956), "Géométrie algébrique et géométrie analytique", Annales de l'Institut Fourier, 6: 1–42, doi:10.5802/aif.59, ISSN 0373-0956, MR 0082175
- Grothendieck, Alexander; Raynaud, Michele (2002). "Revêtements étales et groupe fondamental§XII. Géométrie algébrique et géométrie analytique". Revêtements étales et groupe fondamental (SGA 1) (in French). arXiv:math/0206203. doi:10.1007/BFb0058656. ISBN 978-2-85629-141-2.
- . ISBN 978-4-431-49822-3.
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External links
- Onishchik, A.L. (2001) [1994], "Analytic space", Encyclopedia of Mathematics, EMS Press
- Kiran Kedlaya. 18.726 Algebraic Geometry. Spring 2009. Massachusetts Institute of Technology: MIT OpenCourseWare Creative Commons BY-NC-SA.